Limit of $\frac{x^{6} - 1}{x - 1}$ as $x \to 1$

Q: What is the limit of (x^6 - 1) / (x - 1) as x approaches 1?

The limit of x^6 - 1x - 1 as x → 1 is 6.

The limit of $\frac{x^{6} - 1}{x - 1}$ as $x \to 1$ is $6$ .

\lim_{x \to 1} \frac{x^{6} - 1}{x - 1} = 6

Explain it step by step in chat Make a short video of it

Graph of (x^6 - 1) / (x - 1) near x = 1. The open circle marks the point x = 1, where the function may be undefined; the curve still heads to 6.

How to work it out

1
We want what ( $x^{6}$ - 1) / (x - 1) approaches as x → 1.
$\lim_{x \to 1} \frac{x^{6} - 1}{x - 1}$
2
Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.
3
L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{x^{6} - 1}{x - 1} = \lim \frac{6 x^{5}}{1}$
4
Now substitute x = 1: the denominator is not 0 any more.
$\frac{6}{1} = 6$
5
Method: this is the derivative of $x^{6}$ at 1, namely 6.
6
Result:
$\lim_{x \to 1} \frac{x^{6} - 1}{x - 1} = 6$

LaTeX source of the formulas on this page

\[ \lim_{x \to 1} \frac{x^{6} - 1}{x - 1} = 6 \]
\[ \lim_{x \to 1} \frac{x^{6} - 1}{x - 1} \]
\[ \lim \frac{x^{6} - 1}{x - 1} = \lim \frac{6 x^{5}}{1} \]
\[ \frac{6}{1} = 6 \]

Values of ( $x^{6}$ - 1) / (x - 1) as x → 1 (computed)

x	f(x)	Distance from the limit
1.1	7.7156100	1.72e+0
0.9	4.6855900	1.31e+0
1.01	6.1520151	1.52e-1
0.99	5.8519851	1.48e-1
1.001	6.0150200	1.50e-2
0.999	5.9850200	1.50e-2
1.0001	6.0015002	1.50e-3
0.9999	5.9985002	1.50e-3
1.00001	6.0001500	1.50e-4
0.99999	5.9998500	1.50e-4

Questions

What is the limit of (

x^{6}

- 1) / (x - 1) as x approaches 1?

The limit of $\frac{x^{6} - 1}{x - 1}$ as $x \to 1$ is $6$ .

Which form is it?

0/0.

Limit of x6−1x−1\frac{x^{6} - 1}{x - 1}x−1x6−1​ as x→1x \to 1x→1

How to work it out

Values of (x6x^{6}x6 - 1) / (x - 1) as x → 1 (computed)

Questions

More well-known limits

Limit of $\frac{x^{6} - 1}{x - 1}$ as $x \to 1$

Values of ( $x^{6}$ - 1) / (x - 1) as x → 1 (computed)